TITLE: Evaluate $\int [\cos(\csc^{-1}(\tan(\sec^{-1} (\cot(\sin^{-1}(\sec(\cot^{-1}(\csc(\cos^{-1}(x))))))))))]^2 dx$
QUESTION [5 upvotes]: I was trying to evaluate the below integral:

$$\int \big[\cos(\csc^{-1}(\tan(\sec^{-1}(\cot(\sin^{-1}(\sec(\cot^{-1}(\csc(\cos^{-1}(x))))))))))\big]^2 dx$$

I managed to simplify it  to the below form:
$$\int \frac{3x^2-5}{2x^2-3}dx$$I then got to this form:
$$\frac{3x}{2}+C-\frac{1}{2}\int\frac{1}{2x^2-3}dx$$
I'm stuck at this point; could anyone show me how to proceed?

REPLY [3 votes]: Assuming your simplification is correct (I didn't check), the final step would be handled like this:
$$\int\frac{1}{ax^2 - b}dx = \frac 1b \int\frac{1}{(\sqrt{\frac{a}{b}}x)^2 - 1}dx$$
At this point, you can factor the denominator into $(\sqrt{\frac{a}{b}}x + 1)(\sqrt{\frac{a}{b}}x - 1)$ and use partial fractions.
Alternatively, you could apply the substitution $\sqrt{\frac{a}{b}}x = \cosh y$ which would involve hyperbolic trig functions, if you're comfortable with that.